import random

def random_around_zero(r):
    """ Return a random float in the range [-r, r]"""
    width = 2*r
    return random.random()*width - r

def random_throw(r):
    """Return a random point in the square [(-r,r),(r,r),(r,-r),(-r,-r)]"""
    x = random_around_zero(r)
    y = random_around_zero(r)
    return x, y

def in_circle(r, x, y):
    """Return True if (x,y) is in a circle with radius r centered at 0. 
    Otherwise False."""
    d = (x*x + y*y)**0.5
    return d <= r

def number_in_circle(n_total, r=0.5):
    """Test n_total random points (x,y) in a square with sides 2r and return 
    the number of points that are inside a circle of radius r."""
    n_circle = 0
    for i in xrange(n_total):
        x,y = random_throw(r)
        if in_circle(r, x, y):
            n_circle += 1
    return n_circle

def number_in_circle_monolithic(n_total, r=0.5):
    n_circle = 0
    for i in xrange(n_total):
        x = 2*r*random.random() - r
        y = 2*r*random.random() - r
        d = (x*x + y*y)**0.5
        if d <= r:
            n_circle += 1
    return n_circle

def main():
    for i in range(8):
        n_total = 10**i
        n_circle = number_in_circle(n_total)
        pi = 4.0*n_circle/n_total
        print '1x10^' + str(i) + ':', pi


if __name__ == '__main__':
    main()